It wasn't the obvious hot ticket at this year's fringe. Flight of the Conchords are "the fourth most popular folk parodist act in New Zealand". Their show is tucked under a railway arch on Cowgate, in the dead of night. If they have any posters in Edinburgh, I haven't seen them. And yet Jermaine Clement and Bret McKenzie are this year's buzz comedy act.

It all started with the duo's British debut at last year's fringe. When the stand-ups who preceded them in their Gilded Balloon venue started staying behind to watch Clement and McKenzie, word spread, and "we became a kind of show for other comedians to see after their shows", says McKenzie. "I guess we were quite surprised," says Clement. "They liked it," says McKenzie, "because it was quite different to what they do."

High on Folk, the pair's 2003 offering, comprises about a dozen comedy songs played on acoustic guitar and digital glockenspiel. The musicianship is impressive: Clement and McKenzie's folk-rap crossover, The Hiphopopotamus Meets the Rhymenoceros, sounds like a beatbox Bohemian Rhapsody. And there's more, from Ennio Morricone to acoustic electronica and beyond. There are also blissfully funny lyrics.

It's hard to connect the Conchords in person with the blazing talents on stage. "Kiwis are pretty understated and dry," says McKenzie. This pair are Kiwis through and through: laid-back, unfazed by their success and clueless about what to do with it. They downplay their musical skill. "You can tell when we've learned a new chord," says Clement, "because we'll use it in our next three songs." They didn't even want to play Edinburgh this year. "It costs so much money," says McKenzie. And they think this year's show inferior to its predecessor. "We're surprised," he says, "that people like it so much."

The double-act came together when, as struggling young actors, the two flatmates began learning guitar. "We decided to form a band," says Clement. "It takes ages to learn somebody else's song because you have to remember it all. But if you make up your own, who's gonna pull you up for being wrong?" It is characteristic that their move from music to musical comedy was accidental. "We were supposed to be supplying the music for a comedy night," says McKenzie, "but - and I can't remember how it happened - we ended up being one of the acts."

They are now "one of the major comedy acts in New Zealand", they say. But it doesn't pay their way, and both have other lives: Clement as an actor, McKenzie in non-comic band The Black Seeds. McKenzie also enjoys cult status as the background elf Figwit in Peter Jackson's Lord of the Rings. Although he is on screen for only a handful of seconds, Figwit has scores of websites dedicated to him. "It's because of my elvish looks," McKenzie suggests. He shows me a photograph of two Hebridean dolphins that the Scottish Tolkien Society has sponsored. "One's named after Peter Jackson," Clement tells me, "and one's named after Bret."

So are the pair folk fans? "Only a little bit," says Clement. "I listen to Donovan from time to time." To the Conchords, folk just means acoustic. The joke, says Clement, is that "we're aware that that's not very cool, but we're trying to pretend it's cool". Their act is all about that gap between self-image and reality. They take folk parody deadly seriously. "If we can act as though we're the genuine article, people will find it funnier," says McKenzie. That's why they deliver between-song banter with such hilarious earnestness. "We've been trying to come up with banter that's as boring as possible," says McKenzie. "Last night," giggles Clement, "I think we may have taken that too far."

It's anyone's guess how the Conchords will capitalise on their fringe success. There's a Radio 4 pilot in the offing, about which they profess themselves "excited", albeit in a Kiwi way. "We went to Los Angeles last year," says McKenzie, "and had these really funny interviews with casting directors and people like that. They'd ask us, 'So, what do you guys wanna do?' It was a dream opportunity to say, 'We want to make a film'. And they would have gone, 'Well, here's 20 million'. But we were like, 'Oh, we're not sure. We like going out and watching bands... '"

"It was really exciting," he concludes. "But you needed to have a clear idea of what you wanted to do. And we didn't really have any idea at all."

· At the Gilded Balloon Caves until August 25. Box office: 0131-226 2151.